Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [735,4,Mod(1,735)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(735, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("735.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 735 = 3 \cdot 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 735.a (trivial) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| Self dual: | yes |
| Analytic conductor: | \(43.3664038542\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 35x^{2} + 19x + 196 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} - x^{3} - 35x^{2} + 19x + 196 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 18 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 24\nu + 11 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 18 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 24\beta _1 + 7 \)
|
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−5.10889 | −3.00000 | 18.1008 | 5.00000 | 15.3267 | 0 | −51.6037 | 9.00000 | −25.5444 | ||||||||||||||||||||||||||||||
| 1.2 | −2.36338 | −3.00000 | −2.41442 | 5.00000 | 7.09015 | 0 | 24.6133 | 9.00000 | −11.8169 | |||||||||||||||||||||||||||||||
| 1.3 | 2.92771 | −3.00000 | 0.571487 | 5.00000 | −8.78313 | 0 | −21.7485 | 9.00000 | 14.6386 | |||||||||||||||||||||||||||||||
| 1.4 | 5.54456 | −3.00000 | 22.7422 | 5.00000 | −16.6337 | 0 | 81.7389 | 9.00000 | 27.7228 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(5\) | \(-1\) |
| \(7\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 735.4.a.v | ✓ | 4 |
| 3.b | odd | 2 | 1 | 2205.4.a.bn | 4 | ||
| 7.b | odd | 2 | 1 | 735.4.a.w | yes | 4 | |
| 21.c | even | 2 | 1 | 2205.4.a.bo | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 735.4.a.v | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 735.4.a.w | yes | 4 | 7.b | odd | 2 | 1 | |
| 2205.4.a.bn | 4 | 3.b | odd | 2 | 1 | ||
| 2205.4.a.bo | 4 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(735))\):
|
\( T_{2}^{4} - T_{2}^{3} - 35T_{2}^{2} + 19T_{2} + 196 \)
|
|
\( T_{11}^{4} - 60T_{11}^{3} - 1668T_{11}^{2} + 116640T_{11} - 491680 \)
|
|
\( T_{13}^{4} - 14T_{13}^{3} - 5716T_{13}^{2} + 144360T_{13} - 295776 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - T^{3} + \cdots + 196 \)
|
| $3$ |
\( (T + 3)^{4} \)
|
| $5$ |
\( (T - 5)^{4} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} - 60 T^{3} + \cdots - 491680 \)
|
| $13$ |
\( T^{4} - 14 T^{3} + \cdots - 295776 \)
|
| $17$ |
\( T^{4} - 46 T^{3} + \cdots - 2126592 \)
|
| $19$ |
\( T^{4} - 40 T^{3} + \cdots + 130207104 \)
|
| $23$ |
\( T^{4} - 490 T^{3} + \cdots + 133305088 \)
|
| $29$ |
\( T^{4} + 6 T^{3} + \cdots - 38756736 \)
|
| $31$ |
\( T^{4} - 122 T^{3} + \cdots - 21980448 \)
|
| $37$ |
\( T^{4} - 142 T^{3} + \cdots - 261622368 \)
|
| $41$ |
\( T^{4} - 168 T^{3} + \cdots - 436581360 \)
|
| $43$ |
\( T^{4} - 286 T^{3} + \cdots - 573525504 \)
|
| $47$ |
\( T^{4} + 306 T^{3} + \cdots - 136069632 \)
|
| $53$ |
\( T^{4} + 276 T^{3} + \cdots - 4771872 \)
|
| $59$ |
\( T^{4} + \cdots + 19163570688 \)
|
| $61$ |
\( T^{4} + \cdots - 1312588800 \)
|
| $67$ |
\( T^{4} + \cdots - 21214771200 \)
|
| $71$ |
\( T^{4} + \cdots - 602434684288 \)
|
| $73$ |
\( T^{4} + \cdots + 35615409120 \)
|
| $79$ |
\( T^{4} + \cdots - 140473174016 \)
|
| $83$ |
\( T^{4} + \cdots - 668329496064 \)
|
| $89$ |
\( T^{4} + \cdots + 257087585808 \)
|
| $97$ |
\( T^{4} + \cdots + 891636949152 \)
|
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